# Matematica | DIFFERENTIAL EQUATIONS

## Matematica DIFFERENTIAL EQUATIONS

 0512300025 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2022/2023

 YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2018 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/05 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 LYOUBOMIRA SOFTOVA PALAGACHEVA T
ExamDate
EQUAZIONI DIFFERENZIALI03/04/2023 - 14:00
Objectives
THE MAIN GOAL OF THE COURSE IS TO INTRODUCE THE STUDENTS TO THE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS AND AUTONOMOUS SYSTEMS. THERE ARE PRESENTED DIFFERENT METHODS FOR SOLVING FIRST ORDER AND HIGHER ORDER DIFFERENTIAL EQUATIONS. THE QUALITATIVE STUDY OF AUTONOMOUS SYSTEMS OF THE FIRST ORDER IS ALSO PRESENTED.
Prerequisites
BASIC PROPERTIES OF FUNCTIONS OF A REAL VARIABLE: CONTINUITY, DIFFERENTIABILITY, INTEGRATION. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
Contents
DIRECTIONAL FIELDS, ISOCLINES AND INTEGRAL CURVES. (2H LECTURES)

FIRST-ORDER EQUATIONS RESOLVABLE THROUGH SUBSTITUTIONS, BERNOULLI EQUATION, RICCATI EQUATION, HOMOGENEOUS EQUATIONS AND EXACT DIFFERENTIAL EQUATIONS. EQUATIONS NON-RESOLVABLE WITH RESPECT TO THE DERIVATIVE. (8H LECTURES)

CAUCHY PROBLEM AND DEPENDENCE OF THE SOLUTION FROM THE INITIAL DATA. GRONWALL LEMMA. (2H LECTURES)

HIGHER ORDER EQUATIONS RESOLVABLE THROUGH SUBSTITUTIONS. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS. EULER EQUATION. (8H LECTURES)

FIRST-ORDER EQUATION SYSTEMS. LINEAR SYSTEMS, EXISTENCE AND UNIQUENESS THEOREMS. FUNDAMENTAL MATRIX. EIGENVALUES AND EIGENVECTORS. (8H LECTURES)

AUTONOMOUS SYSTEMS. STABILITY AND BALANCE POINTS. LINEARIZATION OF NONLINEAR SYSTEMS. MATHEMATICAL MODELS. (10H LECTURES)

BOUNDARY PROBLEMS, GREEN FUNCTION. THE STURM-LIOUVILLE PROBLEM. (4H LECTURES)

LAPLACE TRANSFORM AND ANTITRANSFORM. APPLICATION TO DIFFERENTIAL EQUATIONS. (6H LECTURES)
Teaching Methods
FRONTAL LECTURES
Verification of learning
THE LEARNING VERIFICATION TAKES PLACE THROUGH
THE DISCUSSION OF A THESIS CONSISTING OF A THEORY ARGOMENT AND VARIOUS EXERCISES ON THE TOPICS TRATED IN THE COURSE.

AN ORAL EXAM AND INCLUDES A WRITTEN EXAM, TO INTEGRATE THE ORAL EXAM. IN PARTICULAR, ON THE BASIS OF METHODOLOGIES, INSTRUMENTS AND CONTENT GIVEN DURING THE LESSONS, THE STUDENT MUST DEMONSTRATE THAT HE IS ABLE TO UNDERSTAND THE PROBLEM, FIND THE CORRECT MATHEMATICAL-QUANTITATIVE INTERPRETATION, RECOGNIZE THE APPROPRIATE METHOD, UNDERSTAND THE ANSWERS DEDUCED BY THE METHOD AND ITS INFERENCES.
Texts
C.PAGANI, S. SALSA, ANALISI MATEMATICA 2, ZANICHELLI, 2016

A. AMBROSETTI, APPUNTI SULLE EQUAZIONI DIFFERENZIALI ORDINARIE, SPRINGER-VERLAG 2012

V.ARNOLD, ORDINARY DIFFERENTIAL EQUATIONS, SPRINGER-VERLAG, 1992